where \(F_{\mu \nu }\)  is the gauge field strength, \(\psi \)  is the Dirac spinor, \(\gamma ^{\mu }\)  is the Dirac gamma matrix and \(m\)  is the mass of the fermion. The equation of motion for fermion would be the Dirac equation \(i\gamma ^\mu \partial _{\mu } \psi -m \psi = 0\)  [1]}. When the interaction is turned on, the action is modified by a minimum substitution of covariant derivative \(\partial _{\mu } \rightarrow D_{\mu } = \partial _{\mu }+ieA_{\mu }\)  , where \(A_{\mu }\)  is the interaction spin-1 photon gauge field [1]}, [3]}, [4]},
\(S = \int d^4 x \bigg (-\frac{1}{4}F_{\mu \nu }F^{\mu \nu } + i \bar{\psi } \gamma ^\mu D_{\mu }\psi - m \bar{\psi }\psi \bigg ) \,.\) 
